22 1 
the Stability of Ships . 
ASH, equal to the given angle OPQ: when the vessel is inclined 
from the perpendicular through this angle, it will be inter- 
sected by the water’s surface coinciding with the line CH. 
Bisect B C in F, and AH in N ; and join S F, S N : take S I 
to S F, as 2 to 3 ; and S M to S N, in the same proportion. 
Through I and M, draw the lines IK, ML, perpendicular to 
C H. Through the centre of gravity of the vessel G, draw 
G U parallel to C H ; and through the centre of gravity E, of 
the displaced volume BOA, draw E V parallel and equal to 
KL; and in EV take ET to E V, in the same proportion 
which the volume ASH bears to the entire volume displaced 
B OA. Through T, draw T Z perpendicular to G U. G Z is 
the measure of the vessel’s stability. 
To obtain an analytical value of the line GZ, for brevity, 
let the sine of the angle ASH be denoted by s, when radius 
is = i, make sin. HAS = a ; sin. AH S = h ; sin. S C B = c. 
Let G E = d. Also, let the entire volume displaced = V. By 
the rules of trigonometry, it is found that 
* SL== S y X /l. + fZ? I 4 *xy/rrg 
3 V ^ 1 jf T b 
Also SK = — x cos. ASH 4 - sec. ASH. 
3 __ 
The area S C B or AS H = - ng -— 
r __ SA x SB* xtang. ASH / , z i — /j* . 4s x v' 1 — a 2, . 
Wherefore ET == — -xj 4 + s * — + 1 — * b 
+ — x ASH x cos. ASH + sec. ASH. If the breadth BA 
' 6 V ' 
* When the angle SAHr 90, 1 — a z — : o ; and h — cos. S ; in which case, if c is 
put = cos, S, SL = — x 4 + ~T » ^ ut 4 + =4 + tang.* S x sin* S = 4 + 
3 c c __ 
tang.*S— tangASx cos.*Sr=2-fsec.*S-f cos.*S — cos. S 4- sec. S]’. Wherefore SL rz 
S A 
SA 
3 : 
X cos. S + sec. S. 
JG3 81 
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